New materials and the interactions between them are the basis of novel energy storage devices such as supercapacitors and batteries. In recent years, because of the increasing demand for electricity as an energy source, the development of new energy storage materials is among the most actively studied topics. Conductive polymers (CPs), because of their intrinsic electrochemical activity and electrical conductivity, have also been intensively explored. While most of the high capacitance reported in the literature comes from hybrid materials, for example, conductive polymers composed of metal oxides and carbon materials, such as graphene and carbon nanotubes, new chemistry and the 3D structure of conductive polymers remain critical. This comprehensive review focuses on the basic properties of three popular conductive polymers and their composites with carbon materials and metal oxides that have been actively explored as energy storage materials, i.e., polypyrrole (PPy), polyaniline (PANi), and polythiophene (PTh), and various types of electrolytes, including aqueous, organic, quasi-solid, and self-healing electrolytes. Important experimental parameters affecting material property and morphology are also discussed. Electrochemical and analytical techniques frequently employed in material and supercapacitor research are presented. In particular, cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) are discussed in detail, including how to extract data from spectra to calculate key parameters. Pros and cons of CP-based supercapacitors are discussed together with their potential applications.